Interferometry systems and methods

ABSTRACT

In general, in one aspect, the invention features an apparatus that includes an interferometer having a main cavity and an auxiliary reference surface, the main cavity including a partially reflective surface defining a primary reference surface and a test surface. The interferometer is configured to direct a primary portion of input electromagnetic radiation to the main cavity and an auxiliary portion of the input electromagnetic radiation to reflect from the auxiliary reference surface, wherein a first portion of the primary portion in the main cavity reflects from the primary reference surface and a second portion of the primary portion in the main cavity passes through the primary reference surface and reflects from the test surface. The interferometer is further configured to direct the electromagnetic radiation reflected from the test surface, the primary reference surface, and the auxiliary reference surface to a multi-element detector to interfere with one another to form an interference pattern. The auxiliary reference surface is tilted so that the paths of the electromagnetic radiation reflected from the primary reference surface and auxiliary reference are non-parallel at the multi-element detector and the auxiliary reference surface is in the path of the primary portion of the electromagnetic radiation.

CROSS-REFERENCE TO RELATED APPLICATIONS

Under 35 USC § 120, this application is a continuation in partapplication and claims the benefit of U.S. patent application Ser. No.11/079,946, entitled “INTERFEROMETRY SYSTEMS AND METHODS,” filed on Mar.15, 2005, which claims priority under 35 USC §119(e)(1) to ProvisionalPatent Application No. 60/553,312, entitled “METHOD AND APPARATUS FORINTERFEROMETRIC PROFILING WITH REDUCED SENSITIVITY TO ENVIRONMENTALEFFECTS,” filed on Mar. 15, 2004. The entire contents of U.S. patentapplication Ser. No. 11/079,946 and Provisional Patent Application No.60/553,312 are incorporated by reference herein.

BACKGROUND

Interferometric optical techniques are widely used to measure surfaceprofiles of precision optical components.

For example, to measure the surface profile of a test surface, one canuse an interferometer to combine a test wavefront reflected from thetest surface with a reference wavefront reflected from a referencesurface to form an optical interference pattern. Spatial variations inthe intensity profile of the optical interference pattern correspond tophase differences between the combined test and reference wavefrontscaused by variations in the profile of the test surface relative to thereference surface. Phase-shifting interferometry (PSI) can be used toaccurately determine the phase differences and the corresponding profileof the test surface. The surface profile measurement of the test surfaceis relative to the surface profile of the reference surface, which isassumed to be perfect (e.g., flat) or known within the tolerances of themeasurement.

With PSI, the optical interference pattern is recorded for each ofmultiple phase-shifts between the reference and test wavefronts toproduce a series of optical interference patterns that span, forexample, at least a half cycle of optical interference (e.g., fromconstructive, to destructive interference). The optical interferencepatterns define a series of intensity values for each spatial locationof the pattern, wherein each series of intensity values has a sinusoidaldependence on the phase-shifts with a phase-offset equal to the phasedifference between the combined test and reference wavefronts for thatspatial location. Using numerical techniques known in the art, thephase-offset for each spatial location is extracted from the sinusoidaldependence of the intensity values to provide a profile of the testsurface relative the reference surface. Such numerical techniques aregenerally referred to as phase-shifting algorithms.

The phase-shifts in PSI can be produced by changing the optical pathlength from the measurement surface to the interferometer relative tothe optical path length from the reference surface to theinterferometer. For example, the reference surface can be moved relativeto the measurement surface. Alternatively, the phase-shifts can beintroduced for a constant, non-zero optical path difference by changingthe wavelength of the measurement and reference wavefronts. The latterapplication is known as wavelength tuning PSI and is described, e.g., inU.S. Pat. No. 4,594,003 to G. E. Sommargren.

One type of interferometer that is often used for characterizing asurface of a test object is a Fizeau interferometer. In manyembodiments, phase shifting for object surface profiling proceeds bymechanical translation of the reference surface or by wavelength tuning,during which time a computer captures successive frames of aninterference pattern at a detector for later analysis.

In a number of situations, it can be attractive to profile surfacewithout temporal modulation of the Fizeau interference pattern, forexample, to accommodate high-speed measurements of dynamically actuatedparts. Although a variety of such techniques exist for Twymann-Greeninterferometer geometries, including for example spatial phase shiftingor phase shifting based on polarization, these techniques typicallyrequire separating the reference and object beam reflections. However,the common-path characteristics of a large-aperture Fizeauinterferometer can make it difficult to separate the reference andobject beam reflections spatially or by polarization.

SUMMARY

In general, in a first aspect, the invention features an apparatus thatincludes an interferometer having a main cavity and an auxiliaryreference surface, the main cavity including a partially reflectivesurface defining a primary reference surface and a test surface. Theinterferometer is configured to direct a primary portion of inputelectromagnetic radiation to the main cavity and an auxiliary portion ofthe input electromagnetic radiation to reflect from the auxiliaryreference surface, wherein a first portion of the primary portion in themain cavity reflects from the primary reference surface and a secondportion of the primary portion in the main cavity passes through theprimary reference surface and reflects from the test surface. Theinterferometer is further configured to direct the electromagneticradiation reflected from the test surface, the primary referencesurface, and the auxiliary reference surface to a multi-element detectorto interfere with one another to form an interference pattern. Theauxiliary reference surface is tilted so that the paths of theelectromagnetic radiation reflected from the primary reference surfaceand auxiliary reference are non-parallel at the multi-element detectorand the auxiliary reference surface is in the path of the primaryportion of the electromagnetic radiation.

Embodiments of the apparatus may include one or more of the followingfeatures and/or features of other aspects. For example, the auxiliaryreference surface may transmit the primary portion of theelectromagnetic radiation. The main cavity can define a Fizeau cavity.There may be no beam shaping optics in the beam path of second portionbetween the primary reference surface and the test surface.

In general, in another aspect, the invention features an apparatus thatincludes an interferometer having a main cavity and an auxiliaryreference surface, the main cavity including a partially reflectivesurface defining a primary reference surface and a test surface. Theinterferometer is configured to direct input electromagnetic radiationto reflect from the auxiliary reference surface, to reflect from theprimary reference surface and to reflect from the test surface. Theinterferometer is further configured to direct the electromagneticradiation reflected from the test surface, the primary referencesurface, and the auxiliary reference surface to a multi-element detectorto interfere with one another to form an interference pattern. Theauxiliary reference surface is further configured to introduce spatialcarrier fringes in the interference pattern at the multi-elementdetector.

Embodiments of the apparatus may include one or more of the followingfeatures and/or features of other aspects. For example, the auxiliaryreference surface can be positioned in the path of the inputelectromagnetic radiation directed to the primary reference surface.

The apparatus can include a beam splitter configured to split the inputelectromagnetic radiation into an auxiliary portion and a primaryportion, wherein the beam splitter directs the auxiliary portion along apath to the auxiliary reference surface and the beam splitter directsthe primary portion along a path to the primary reference surface.

The auxiliary reference surface can be a partially transmissive surface.Alternatively, the auxiliary reference surface can be a reflectivesurface. The auxiliary reference surface can be a planar surface.Alternatively, or additionally, the primary reference surface can be aplanar surface. In embodiments where both the primary and auxiliaryreference surfaces are planar surfaces, the auxiliary reference surfaceand the primary reference surface can be non-parallel.

In some embodiments, the primary reference surface and the auxiliaryreference surface are opposing surfaces of an optical component. Theoptical component can be a transmissive wedge.

The main cavity can define a Fizeau cavity.

The auxiliary reference surface can be configured to transmit the inputelectromagnetic radiation directed to reflect from the primary referencesurface. The primary reference surface can be configured to reflectinput electromagnetic radiation along a first path and the auxiliaryreference surface is configured to reflect input electromagneticradiation along a second path, where the second path overlaps the firstpath and is non-parallel to the first path.

In some embodiments, there are no beam shaping optics in the beam pathof second portion between the primary reference surface and the testsurface.

The apparatus can include a means for selectively preventingelectromagnetic radiation from the test surface from reaching thedetector.

The apparatus can include the multi-element detector and an electroniccontroller, wherein the electronic controller is configured to determinesurface profile information about the test surface based on theinterference pattern. The electronic controller can be configured todetermine surface profile information about the test surface based onthe interference pattern formed by the electromagnetic radiationreflected from the test surface, the primary reference surface, and theauxiliary reference, and a second interference pattern formed byelectromagnetic radiation reflected from the primary reference surfaceand the auxiliary reference, with no electromagnetic radiation reflectedfrom the test surface reaching the detector.

The auxiliary reference surface can be tilted relative to an opticalaxis of the interferometer to form the spatial carrier fringes in theinterference pattern. The apparatus can include a quadrature phasedetection system including the multi-element detector. Theinterferometer can include a fold optic configured to allow the primaryreference surface to be upward facing and part of a mount configured tosupport the test surface. In some embodiments, the auxiliary referenceis coupled to a transducer configured to vary an optical path length ofelectromagnetic radiation reflected from the auxiliary reference to thedetector.

In general, in a further aspect, the invention features an apparatusthat includes a multi-element detector, an interferometer comprising aprimary reference surface and a test surface, the interferometer beingconfigured to interfere electromagnetic radiation reflected from theprimary reference surface and electromagnetic radiation reflected fromthe test surface to form an interference pattern at the multi-elementdetector, and a means for generating spatial carrier fringes in theinterference pattern at the multi-element detector.

Embodiments of the apparatus may include one or more of the followingfeatures and/or features of other aspects. For example, the apparatuscan further include an electronic controller configured to determinesurface profile information about the test surface based on theinterference pattern and the carrier fringes. The means for generatingthe carrier fringes can include an auxiliary reference mirror configuredto direct auxiliary electromagnetic radiation along a path to themulti-element detector, wherein the auxiliary electromagnetic radiationinterferes with the electromagnetic radiation reflected from the primaryreference surface and the test surface to form the carrier fringes inthe interference pattern. The interferometer can be a Fizeauinterferometer.

In general, in another aspect, the invention features apparatus thatinclude an interferometer having a main cavity and an auxiliaryreference surface, the main cavity including a primary reference surfaceand a test surface. The interferometer is configured to direct a primaryportion of input electromagnetic radiation to the main cavity and anauxiliary portion of the input electromagnetic radiation to reflect fromthe auxiliary reference surface, wherein a first portion of the primaryportion in the main cavity reflects from the primary reference surfaceand a second portion of the primary portion in the main cavity reflectsfrom the test surface. The interferometer is further configured todirect the electromagnetic radiation reflected from the test surface,the primary reference surface, and the auxiliary reference to amulti-element detector to interfere with one another to form aninterference pattern.

In general, in another aspect, the invention features methods thatinclude directing a primary portion of input electromagnetic radiationto a main cavity of an interferometer, wherein a first portion of theprimary portion in the main cavity reflects from a primary referencesurface of the main cavity and a second portion of the primary portionin the main cavity reflects from a test surface. The methods furtherinclude directing an auxiliary portion of the input electromagneticradiation to reflect from an auxiliary reference surface of theinterferometer, and directing the electromagnetic radiation reflectedfrom the test surface, the primary reference surface, and the auxiliaryreference surface to a multi-element detector to interfere with oneanother forming an interference pattern.

Embodiments of the apparatus and/or methods can include one or more ofthe following features.

The methods can include determining surface profile information aboutthe test surface based on the interference pattern formed by theelectromagnetic radiation reflected from the test surface, the primaryreference surface, and the auxiliary reference, and a secondinterference pattern formed by electromagnetic radiation reflected fromthe primary reference surface and the auxiliary reference, with noelectromagnetic radiation reflected from the test surface reaching thedetector.

In embodiments, the primary reference surface can be a partiallyreflective surface. In some embodiments, the second portion of theprimary portion in the main cavity passes through the primary referencesurface and reflects from the test surface. The main cavity can define aFizeau cavity. There can be beam shaping optics in the beam path ofsecond portion between the primary reference surface and the testsurface.

The surface area of the auxiliary reference surface can be smaller thanthat of the primary reference surface. The auxiliary reference surfacecan be flat and the primary reference surface can be curved. Theapparatus can further include a means for selectively preventingelectromagnetic radiation from the test surface from reaching thedetector. For example, the apparatus can include an aperture between theprimary reference surface and the test surface.

The apparatus can further include the multi-element detector and anelectronic controller, wherein the electronic controller is configuredto determine surface profile information about the test surface based onthe interference pattern. The electronic controller can be configured todetermine surface profile information about the test surface based onthe interference pattern formed by the electromagnetic radiationreflected from the test surface, the primary reference surface, and theauxiliary reference, and a second interference pattern formed byelectromagnetic radiation reflected from the primary reference surfaceand the auxiliary reference, with electromagnetic radiation from thetest surface being prevented from the reaching the detector. Theauxiliary reference surface can be tilted relative to an optical axis ofthe interferometer to form spatial carrier fringes in the interferencepattern. The apparatus can include a quadrature phase detection systemincluding the multi-element detector. In some embodiments, the auxiliaryreference is mounted on a transducer configured to vary an optical pathlength to the multi-element detector for electromagnetic radiationreflected from the auxiliary reference. The interferometer can include afold optic (e.g., a mirror or a prism) configured to allow the primaryreference surface to be upward facing and part of a mount configured tosupport the test surface. The interferometer can include a beam splitter(e.g., a non-polarizing beam splitter or a polarizing beam splitter) forseparating the primary portion of the input electromagnetic radiationfrom the auxiliary portion of the input electromagnetic radiation. Theinterferometer can include one or more imaging optics for imaging thetest surface onto the multi-element detector. The apparatus can furtherinclude a source for the input electromagnetic radiation (e.g., alaser).

The interferometer can be a Fizeau interferometer, a Michelsoninterferometer, a Linnik interferometer, or a Mirau interferometer.

In certain aspects, the methods are implemented by an embodiment of theapparatus.

Among other advantages, embodiments of the apparatus and methods canprovide Fizeau interferometers that are mechanically stable andrelatively insensitive to environmental sources of uncertainty, such asvibrations, which can cause surfaces in the interferometer to move whilethe interferometer is being used to make measurements on a test part, orbetween testing different parts. The stability and relativeinsensitivity to environmental effects result in interferometry systemsthat are reliable and accurate.

Furthermore, embodiments include interferometry systems than can performsurface profiling measurements based on a single frame of aninterference pattern, rather than multiple frames that are commonlyrequired in systems that use phase shifting techniques. Single framemeasurements can be performed more rapidly than multiple frametechniques. Furthermore, single frame techniques can eliminate the needfor moving parts (e.g., transducers for phase shifting), therebyreducing the cost of interferometry systems.

Single frame measurements using Fizeau interferometers can beimplemented without any optical components between the reference surfaceand test surface. Accordingly, sources of error in measurements madeusing the Fizeau interferometers are reduced compared to systems thatinclude optical components in the measurement beam path.

Aspects of the invention can be implemented using relatively simplealterations to commercially available interferometry systems. Forexample, commercially available Fizeau interferometer systems can bereadily adapted to include an auxiliary mirror using relatively fewadditional components, and those components can be relativelyinexpensive.

In some embodiments, the apparatus and methods can include mechanicalphase-shifting interferometer systems having improved propertiescompared to conventional phase-shifting systems. For example, Fizeauinterferometer systems can include an auxiliary mirror mounted on atransducer, so that the mechanical phase shifting is performed outsideof the Fizeau cavity. In large aperture systems, this can beparticularly advantageous since the auxiliary mirror can be much smallerin size than the reference mirror, requiring a less complicatedtransducer system for phase shifting.

Including an auxiliary mirror in a Fizeau interferometer system canallow for relatively easy implementation of quadrature phasemeasurements using polarization. This allows beam polarization to bemanaged way from the Fizeau cavity, allowing such phase measurements tobe made without including any additional components (e.g., a quarterwave plate) between the reference and test surfaces.

Auxiliary mirrors also provide a convenient way to measure phases usingcarrier fringe techniques. In certain embodiments, for example, carrierfringes can be introduced into an interference pattern generated using aFizeau interferometer by tilting the auxiliary mirror relative to thebeam path, rather than the reference mirror. Tilting the auxiliarymirror also allows one to make measurements with the reference mirrororientated on the null position.

The details of one or more embodiments of the invention are set forth inthe accompanying drawings and the description below. Other features,objects, and advantages of the invention will be apparent from thedescription and drawings, and from the claims.

DESCRIPTION OF DRAWINGS

FIG. 1 is a diagram showing an embodiment of an interferometry assemblythat includes a Fizeau interferometer augmented by an auxiliaryreference mirror.

FIG. 2 is a diagram showing the interferometry assembly from FIG. 1modified to have a narrow annular aperture, blocking the view of theobject and therefore preserving a portion of the single-surfacereflection from the Fizeau reference.

FIG. 3 is a diagram showing an embodiment of an interferometry assemblythat includes a spherical Fizeau cavity and an auxiliary referencemirror.

FIG. 4 is a diagram showing an embodiment of an interferometry assemblythat includes a Fizeau interferometer augmented by an auxiliaryreference mirror configured for instantaneous quadrature phasemeasurement using polarization.

FIG. 5 is a diagram showing an embodiment of an interferometry assemblythat includes a Fizeau interferometer augmented by an auxiliaryreference mirror configured for phase measurement using mechanical phaseshifting of the auxiliary reference mirror.

FIG. 6 is a diagram showing an embodiment of an interferometry assemblythat includes an upward-looking Fizeau interferometer augmented by anauxiliary reference.

FIG. 7 is a diagram showing another embodiment of an interferometryassembly that includes an upward-looking Fizeau interferometer augmentedby an auxiliary reference.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

Embodiments described below include Fizeau interferometers that areaugmented by the inclusion of an auxiliary reference mirror. Theauxiliary reference mirrors allow a variety of interferometry techniquesto be implemented using a Fizeau interferometer without having tointroduce any additional components between the Fizeau's reference andtest surfaces or altering the alignment between these surfaces. Forexample, the auxiliary mirror can be configured to introduce carrierfringes into a Fizeau interference pattern, enabling characterization ofa test surface from a single interferogram frame formed using the testsurface. Furthermore, as another example, the auxiliary mirror enablesquadrature phase measurements using polarization to be implemented in aFizeau interferometer relatively easily and without requiring additionaloptical components between the reference and test surfaces of the Fizeauinterferometer. Since any optical components in the measurement beampath are sources of first order errors in measurements made using aFizeau interferometer, using an auxiliary mirror can allow thesetechniques to be implemented without any substantial loss in systemaccuracy.

Auxiliary reference mirrors can also be used to implement mechanicalphase shifting in a Fizeau interferometer without moving either of thereference or test surfaces. The environmental stability associated withthe stationary Fizeau cavity can enhance the system's accuracy andrepeatability.

Referring to FIG. 1, interferometry system 100 is adapted to measure theoptical interference produced by reflections from a reference surface141 of a reference optic 140, a test surface 191 of a test object 190(e.g., an optical flat), and a surface 151 of an auxiliary referencemirror 150, which may be substantially smaller than reference optic 140.Reference surface 141 and test surface 191 define a Fizeau cavity,labeled as cavity 101 in FIG. 1. System 100 includes a mount 192 forsupporting test object 190 relative to reference optic 150.

Interferometry system 100 also includes a beam splitter 120 thatseparates a beam from a light source (e.g., a laser diode, HeNe laser orthe like) into two component beams, corresponding to a reflectedcomponent and a transmitted components of the input beam. The reflectedcomponent is directed towards Fizeau cavity 101, while the transmittedcomponents is directed towards auxiliary reference mirror 150. Beamsplitter 120 also combines light reflected from Fizeau cavity 101 andfrom auxiliary reference mirror 150, and directs the combined light beamto a pixelated detector 160 (e.g., a CCD camera).

As depicted in FIG. 1, interferometry system 100 also includes severaloptical components, including a collimating lens 115 that collimatesdiverging light from source 110 before the light is incident on beamsplitter 120. Interferometry system 100 also includes beam shapingoptics, namely lenses 130 and 132, which expand the light reflected bybeam splitter 120 prior to contacting reference optic 140. In general,interferometry systems 100 can include further optical components inaddition to collimating lens 115 and beam shaping lenses 130 and 132.

During operation, source 110 illuminates beam splitter 120, whichtransmits a portion of the illumination to reflect from auxiliaryreference mirror surface 151, and reflects a portion of the illuminationtowards Fizeau cavity 101. This illumination is partly transmitted byreference optic 140 and reflects from surface 191 of test object 190. Inaddition, a portion of the illumination incident on reference optic 140from beam splitter 120 is reflected by reference surface 141.Illumination reflected from reference surface 141 and from test objectsurface 191 propagate along a common path from Fizeau cavity 101 backthrough beam splitter 120 onto detector 160. In addition, beam splitterreflects a portion of illumination reflected from surface 151 ofauxiliary reference mirror 150 towards detector 160. Wavefronts incidenton detector 160 from mirror 150, reference surface 141, and test objectsurface 191 interfere, generating a pattern of fringes of varyingintensity.

Interferometry system 100 includes an electronic controller 180, whichis in communication with detector 160. Electronic controller 180includes a frame grabber for storing images detected by detector 160.Electronic controller 180 analyzes the images stored by the framegrabber, and provides a user with information about test surface 191based on the analysis.

Auxiliary reference mirror 150 is nominally flat and is oriented so thatmirror surface 151 is tilted with respect to the path of illuminationreflected from reference surface 141 to introduce carrier fringes to theinterference pattern formed by the measurement and reference wavefrontsat detector 160. In general, the tilt is sufficient to introducemultiple fringes into an interference pattern formed by wavefrontsreflected from auxiliary reference mirror 150 and wavefronts reflectedfrom reference surface 141. In some embodiments, auxiliary referencemirror 150 is oriented so that the carrier fringes have a periodcorresponding to about three or more detector pixels at detector 160(e.g., about five or more, about eight or more, about 10 or more, about20 or more detector pixels). Auxiliary reference mirror 150 can bemounted on an adjustable mount to facilitate easy adjustment of theauxiliary mirror's orientation.

Determining a surface profile of test surface 191 using interferometrysystem 100 involves a two-step process: one interferometer measurementwithout the test object 191 and one measurement with the test object.For each measurement, electronic controller 180 determines aninterference phase, θ, and a signal modulation, M, for the image data ateach detector pixel.

One way to extract phase information from each measurement is to performa spatial Fourier transform of the interference pattern and subsequentlyidentify the carrier fringe frequency with a digital filter in thefrequency domain. An inverse transform of the filtered spectrum providesthe phase θ and signal modulation M information at each pixel. Phaseextraction methods using Fourier transforms are described, for example,by W. Macy in “Two Dimensional Fringe Pattern Analysis,” Appl. Opt., 22,pp. 3898-3901 (1983), the entire contents of which are herebyincorporated by reference.

The surface height profile for test surface 141 is related to the phaseinformation for each measurement based on the following derivation. InFIG. 1, when test optic 140 is not included in the system, the systemmeasures the complex reflectivity r₂t² with respect to r₁, where r₁ isthe effective amplitude reflectivity of the auxiliary reference, r₂ isthe internal amplitude reflectivity of reference surface 141, and t isthe effect of transmission through the beam shaping optics and throughthe substrate of the reference optic 140.

Setting aside irrelevant overall constants related to cavity length andphase change on reflection effects, the measured phase at each pixel isgiven byθ_(A)=θ′₂−θ₁  (1)whereθ₁=arg(r ₁)  (2)θ′₂=arg(t ² r ₂)  (3)Herer ₁ =√{square root over (R ¹ )} exp( iθ ₁)tm (4)r ₂ =√{square root over (R ² )} exp( iθ ₂)  (5)t=√{square root over (T)}exp(iθ _(t))  (6)and thereforeθ′₂=2θ_(t)+θ₂  (7)The phase values relate to surface heights according toθ₁=2kh ₁  (8)θ₂=2kh ₂  (9)fork=2π/λ  (10)and λ is the source wavelength.

The measured signal modulation can be expressed as follows:M _(A)=2VT√{square root over (R ¹ R ² )}  (11)where V is a fringe visibility coefficient, assumed to be independent offield position and constant over time.

For the interference pattern acquired with test object 190 included, theray tracing is common path for light reflected from test surface 191 andreference surface 141, with a phase measurement for each pixel now givenbyθ_(B)=θ′_(Z)−θ₁  (12)whereθ′_(Z)=2θ_(t)+θ_(Z)  (13)and the phase θ_(Z) returned by the Fizeau cavity itself isθ_(Z)=arg(z)  (14)Forz=(r ₂ +r ₃)  (15)The measured signal modulation isM _(B)=2V T√{square root over (R ¹ Z)}  (16)where z=|z|².

The phase term, θ₃, corresponding to the surface profile of test surface191 at each pixel, is determined from the measured phases and signalmodulations as follows. The complex reflectivity, z, of Fizeau cavity101 can be calculated for each pixel from the measured data from:$\begin{matrix}{z = {\frac{M_{B}}{2{VT}\sqrt{R_{1}}}{{\exp\quad\left\lbrack {{\mathbb{i}}\left( {\theta_{B} - \theta_{A} + \theta_{2}} \right)} \right\rbrack}.}}} & (17)\end{matrix}$The reflectivity of reference surface 141 is from Eq. (11) and Eq.(5)$\begin{matrix}{r_{2} = {\frac{M_{A}}{2{VT}\sqrt{R_{1}}}\exp\quad{\left( {{\mathbb{i}}\quad\theta_{2}} \right).}}} & (18)\end{matrix}$Therefore, since from Eq. (15) $\begin{matrix}{{r_{3} = {z - r_{2}}},{{one}\quad{has}}} & (19) \\{r_{3} = {{\frac{\exp\quad\left( {{\mathbb{i}}\quad\theta_{2}} \right)}{2{VT}\sqrt{R_{1}}}\left\lbrack {{M_{B}\exp\quad\left( {{{\mathbb{i}}\quad\theta_{B}} - {{\mathbb{i}}\quad\theta_{A}}} \right)} - M_{A}} \right\rbrack}.}} & (20)\end{matrix}$the complex argument of which is the desired phase: $\begin{matrix}{{\theta_{\quad 3} = {{\arg\left\lbrack {{M_{\quad B}\exp\left( {{\mathbb{i}\theta}_{\quad B} - {\mathbb{i}\theta}_{\quad A}} \right)} - M_{\quad A}} \right\rbrack} + \theta_{\quad 2}}}.} & (21)\end{matrix}$(21)

Where the profile of reference surface, h₂, is known, equation (21)allows one to determine the surface profile of test surface 191independent of the effects t of the optical system.

In certain embodiments, it may be desirable to reduce the reflectivityof reference surface 141. In such instances, an antireflection (AR)coating can be applied to reference surface 141. This reduces thefinesse of Fizeau cavity 101, which can reduce the effect of errorsassociated with instability of the optical path between referencesurface 141 and auxiliary mirror surface 151. For example, the techniqueas described above is sensitive to the drift of the “internal” cavityformed by reference surface 141 and auxiliary mirror reference surface151 than may occur between the first and the second measurements,potentially introducing measurement errors in unstable environments. Thesensitivity is generally greatest for high-finesse fringes between r₂and r₃. The magnitude of the print through declines with the finesse ofFizeau cavity 101, which is reduced when an AR coating is applied toreference surface 141. For example, if R₂=0.5% and R₃=4%, the resultingerrors for an internal cavity (i.e., the cavity formed by referencesurface 141 and auxiliary mirror surface 151) drift are reduced by afactor of three.

Alternatively, or additionally, interferometer system 100 can bedesigned to be sufficiently rigid to substantially prevent any internalcavity drift between first and second measurements associated withcharacterizing a part.

In certain embodiments, the stability of the internal cavity can bemonitored between measurements by including a portion of theinterferometer system that monitors the interference pattern associatedwith the internal cavity regardless of whether or not a test object ispositioned in the system. For example, an annulus can be provided in thefield of view of detector 160 that examines only reference surface 141,and excludes the test surface 191 reflection, r₃.

Referring to FIG. 2, in some embodiments, an annulus can be provided byincluding an aperture 210 within Fizeau cavity 101. For example, if testobject 190 is larger than the system's field of view, aperture 210excludes reflections from test surface 151 by masking portions of thetest surface.

Including such an annulus in interferometer 100 allows one to determinethe relative tip, tilt and piston of the internal cavity every time ameasurement is made. The measured drift values modify the Eq. (21) toθ₃=arg[M _(B) exp(iθ _(B) −iθ _(A) +iθ _(drift))−M _(A)]+θ₂  (22)where θ_(drift) is the phase change at a pixel associated with the driftin the internal cavity between measurements.

Referring to FIG. 3, an interferometry system 300 can be adapted forprofiling a curved test surface of a test object 390. Interferometrysystem 300 includes a reference object 310 that has a curved (e.g.,spherical) reference surface 311. Reference object 310 and test object390 defines a Fizeau cavity 301. The interferometry system's beamshaping optics also includes an additional lens 320 which focusesillumination from beam splitter 120 so that it is nominally normallyincident on the curved surface of test object 390. Aside from thedifferent geometry of the Fizeau cavity, the operation of interferometrysystem 300 is the same as interferometry system 100.

While in the preceding embodiments phase information is determined usingcarrier fringes, in general, other techniques can also be used. Forexample, in some embodiments phase measurements can be obtained from asingle interference pattern using polarization to provide phasequadrature between orthogonal polarizations in the interference pattern.Referring to FIG. 4, an example of an interferometry system adapted forphase quadrature measurements using polarization is interferometrysystem 400. System 400 includes a polarizing beam splitter (PBS) 420which splits incident illumination from source 110 (not shown in FIG. 4)into component beams having orthogonal linear polarization states. Onebeam reflects from surface 151 of auxiliary reference mirror 150, whilethe other reflects from the reference and test surface of Fizeau cavity101. Quarter wave plates 410 are positioned in the path of each beam sothat the polarization state of each beam is rotated by 90 degrees. Thisensures that the beam originally transmitted by PBS 420 is nowreflected, and the originally reflected beam, now reflected from Fizeaucavity 101, is transmitted by the PBS.

PBS 420 directs illumination reflected from auxiliary reference mirror150 and Fizeau cavity 101 towards a detector assembly 401 that includespixelated detectors 460 and 470, and non-polarizing beam splitter 435.Detector assembly 401 also includes lenses 430, 440, and 445, whichserve to focus and collimate the illumination from PBS 420. Beamsplitter 435 reflects a portion of the incoming illumination from PBS420 toward detector 470, and transmits a portion towards detector 460. Alinear polarizer is positioned between each detector and beam splitter435. In particular, polarizer 462 is positioned in front of detector460, and polarizer 472 is positioned in front of detector 470. Bothpolarizers are oriented at 45 degrees with respect to the transmissionaxis of PBS 420, ensuring that both detectors sample illuminationreflected from both the auxiliary reference mirror and the Fizeaucavity. A quarter waveplate is positioned between polarizer 472 and beamsplitter 435, and is oriented to introduce a 90 degree phase shift intothe polarization state transmitted by polarizer 472. Accordingly, theinterference phase of the interference pattern at each detector pixelfor one detector is offset by 90 degrees relative to the phase of theinterference pattern at the corresponding pixel of the other detector.

Signals in quadrature allow for rapid measurement of modulation andphase assuming that the constant intensity offset I^(DC) is known. Thusif the measured intensity without the object for one camera isg _(A) =I _(A) ^(DC) +M _(A) cos(θ_(A))  (23)then for the other camera in quadrature the intensity is $\begin{matrix}{{g_{A}^{quad} = {I_{A}^{DC} + {M_{A}\sin\quad\left( \theta_{A} \right)}}}{then}} & (24) \\{{\tan\quad\left( \theta_{A} \right)} = \frac{g_{A}^{quad} - I_{A}^{DC}}{g_{A} - I_{A}^{DC}}} & (25) \\{M_{A}^{2} = {\left( {g_{A}^{quad} - I_{A}^{DC}} \right)^{2} + \left( {g_{A} - I_{A}^{DC}} \right)^{2}}} & (26)\end{matrix}$To determine I_(A) ^(DC), one can make a separate measurement with theauxiliary reference either highly tilted or in rapid movement so as toaverage out the modulation terms M_(A) cos(θ_(A)) and M_(A) sin(θ_(A)).In an alternative embodiment, the addition of at least one more camerawith an additional phase shift provides sufficient information to solvefor modulation, average intensity and phase simultaneously, as describedby R. Smythe and R. Moore, Proc. Soc. Phot. Opt. Eng. 429, 16 (1983).

In further embodiments, polarization-encoded reference and measurementbeams project multiple images at various phase shifts onto a singlecamera detector, in a manner similar to that described by Kujawinska etal. in the Chapter “Spatial phase measurement methods,” in InterferogramAnalysis: Digital Fringe Pattern Measurement Techniques, edited by DavidW. Robinson and Graeme T. Reid, Institute of Physics Publishing,Philadelphia, Pa. (May 1, 1993), the entire contents of which are herebyincorporated by reference, and by James Millerd, et al., in “PixelatedPhase-Mask Dynamic Interferometer,” Proc. SPIE, Vol.5531 (2004), theentire contents of which are hereby incorporated by reference.

In some embodiments, mechanical phase shifting can be used to determinephase information in interferometry systems that include auxiliarymirrors. For example, referring to FIG. 5, in an interferometry system500, auxiliary mirror 150 is mounted on a transducer 510 (e.g., apiezoelectric transducer) which varies the optical path length betweenauxiliary reference surface 151 and beam splitter 120. Transducer 510 iscontrolled by signals from electronic controller 180, which synchronizesdata acquisition at detector 160 with the motion of the auxiliaryreference mirror. Transducer 510 displaces auxiliary reference mirror150 smoothly over a range of approximately one wavelength, therebyintroducing a sequence of N phase shift increments α while detector 160records corresponding N intensity values g_(j) for j=0 . . . N−1 foreach image pixel, as described, e.g., in Encyclopedia of Optics, vol. 3(2004, Wiley-VCH Publishers, Weinheim) pp. 2100-2101. For example, forN=7 and phase shift increments α=π/2, the phase is given by$\begin{matrix}{{\tan\quad(\theta)} = \frac{{7\left( {g_{2} - g_{4}} \right)} - \left( {g_{0} - g_{6}} \right)}{{{- 4}\left( {g_{1} + g_{5}} \right)} + {8g_{3}}}} & (27)\end{matrix}$and the signal modulation is given by $\begin{matrix}{M^{2} = \frac{\left\lbrack {{7\left( {g_{2} - g_{4}} \right)} - \left( {g_{0}g_{6}} \right)} \right\rbrack^{2} + \left\lbrack {{8g_{3}} - {4\left( {g_{1} + g_{5}} \right)}} \right\rbrack^{2}}{16^{2}}} & (28)\end{matrix}$Alternatively, or additionally, other methods of determininginterference phase and modulation with the aid of an auxiliary referencemirror can be used.

In the embodiments described above, the aperture of the Fizeau cavity issufficiently large to capture reflections from the entire test surface.This is achieved, in part, by using a large reference optic (e.g., thesize of reference surface 141 is the same size or larger than the sizeof test surface 191). Indeed, where an interferometry system is designedfor characterizing large test parts, the corresponding reference opticcan be large and heavy. Mechanical phase shifting by translating thesmall auxiliary mirror can be advantageous since the size of the Fizeauaperture does not place the same size requirements on the auxiliarymirror as it does for the reference optic 140. Accordingly, thetransducer can be smaller and simpler compared to a transducer forreference optic 140. Furthermore, due to the physical distance betweenthe auxiliary reference and the Fizeau cavity, the auxiliary referencemirror can be physically isolated more effectively from the test objectcompared to reference object 140, providing better stability of theFizeau cavity during data acquisition.

Referring to FIG. 6, in some embodiments, interferometry systems can beconfigured to be upward-looking, for easy test part handling.Interferometry system 600, for example, includes a fold mirror 610between beam splitter 120 and the reference object 640 and test part690. fold mirror 610 directs illumination from beam splitter 120 along avertical path to illuminate reference optic 640 which lies in ahorizontal plane. Test part 690 can be easily mounted with test surface691 relative to reference surface 641 by laying the test parthorizontally on a mount 692 that maintains a small amount of separationbetween test surface 691 and reference surface 641. Since gravityautomatically aligns the test part to reference object 640, tip and tiltalignments can be eliminated for all flat parts, regardless of size, inthis configuration.

In the foregoing embodiments, the path of light between to the auxiliaryreference surface and the primary reference surface are located alongdifferent paths. More generally, however, other configurations can alsobe used. In some embodiments, an interferometry system can include anauxiliary reference surface that is positioned in the path of light fromthe source to reference surface. For example, referring to FIG. 7, aninterferometry system 700 includes a transmissive optical component 720whose opposing surfaces 730 and 740 serve as the primary referencesurface and auxiliary reference surface, respectively. A test object 710is positioned with a test surface 711 relative to primary referencesurface 730.

During operation, auxiliary reference surface 740 reflects a portion ofthe light from source 110 incident thereon. Primary reference surface730 reflects a portion of the light transmitted by the auxiliaryreference surface. The light transmitted by transmissive opticalcomponent 720 is incident on test surface 711 and at least a portionreflects therefrom. Light reflected from the primary reference surface730 and test surface 711 is transmitted again by transmissive opticalcomponent 720 and overlaps with the light reflected from auxiliaryreference surface 740. The overlapping light results in an interferencepattern at detector 160.

Transmissive optical component 720 is a wedge element. The wedge elementtransmits at least some of the incident light from the source. Forexample, the wedge element can transmit about 50% or more (e.g., about60% or more, about 70% or more, about 80% or more) of light at thesystem's operational wavelength normally incident thereon.

The wedge angle of transmissive optical component 720 can vary asdesired. Typically, the wedge angle is selected based on the desiredcarrier fringe frequency in the detected interference pattern. The wedgeangle should be sufficiently small so that the light reflected fromauxiliary reference surface overlaps the light from primary referencesurface at detector 160. Typically, the wedge angle is selected tointroduce an appropriate number of carrier fringes at the detector,consistent with the detector resolution. In some embodiments, the wedgeangle is in a range from 0.1° to about 10°.

Auxiliary reference surface 740 may be the same general size and shapeas primary reference surface 730.

In general, the thickness of transmissive optical component 720 can varyas desired. In certain embodiments, the thickness of the component canbe selected so that the optical path length between the auxiliaryreference surface is a certain amount (e.g., different from the opticalpath length of the primary cavity). In the present embodiments, thethickness of transmissive optical component 720 determines the distancebetween the auxiliary reference surface and the primary referencesurface. In certain embodiments, auxiliary reference surface 740 ispositioned relatively close to primary reference surface 730. Forexample, auxiliary reference surface 740 can be about 5 cm or less(e.g., about 3 cm or less, about 2 cm or less, about 1 cm or less, about0.5 cm or less) from primary reference surface 730.

The reflectivity of auxiliary reference surface 740 can vary as desired.Generally, transmissive optical component 720 is designed so thatauxiliary reference surface 740 has a reflectivity that providessuitable contrast of the carrier fringes in the interference pattern atthe detector. Typically, the reflectivity of auxiliary reference surface740 is the same as or greater than the reflectivity of primary referencesurface 730. In some cases, the reflectivity of the auxiliary andprimary reference surfaces are relatively high (e.g., about 50% ormore). Alternatively, their reflectivities can be relatively low (e.g.,about 10% or less). In certain embodiments, the reflectivity ofauxiliary reference surface 740 is relatively high, while thereflectivity of primary reference surface 730 is relatively low. Themeasurement can become increasingly sensitive to changes in the distancebetween the auxiliary and primary reference surfaces as theirreflectivities become closer. Accordingly, in certain embodiments, itmay be advantageous for their reflectivities to be different.

In some embodiments, one or both of auxiliary reference surface 740 andprimary reference surface 730 can include an optical coating, such as acoating that decreases the reflectivity of the surface (e.g., anantireflection coating) or a coating that increases the reflectivity ofthe surface (e.g., a high-index coating).

Although auxiliary reference surface 740 and primary reference surface730 are opposing surfaces of a common component in interferometry system700, in general, the auxiliary reference surface and the primaryreference surface can be surfaces of different components. For example,the auxiliary reference surface can be a surface of a first transmissiveoptical flat (e.g., a glass flat with parallel opposing surfaces) whilethe primary reference surface is a surface of a second transmissiveoptical flat where the flats are oriented so that their surfaces arenon-parallel.

The component that includes the auxiliary reference surface can bemounted in an adjustable mount (e.g., manually adjustable or adjustableusing an electronically controlled actuator). For example, the mount canbe used to adjust the distance between the auxiliary reference surfaceand the primary reference surface. Alternatively, or additionally, themount can be used to adjust the orientation of the auxiliary referencesurface with respect to the primary reference surface. In someembodiments, the mount can include piezoelectric actuators that areconfigured to effect phase shifting of the carrier fringes, enablingphase shifting techniques to be used to profile the surface of the testobject.

While the foregoing embodiments involve the use of an auxiliary mirrorin conjunction with a Fizeau interferometer, in general, the phase andmagnitude techniques using an auxiliary mirror can be used with othertypes of interferometers as well. For example, the phase and magnitudetechniques using an auxiliary mirror can be adapted for use with aconventional interference microscope objective, such as a Mirau,Michelson or Linnik. Conventional interference microscopes can beadapted to include an auxiliary reference mirror with relatively littlemodification of the standard geometry.

A number of embodiments of the invention have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the invention.Accordingly, other embodiments are within the scope of the followingclaims.

1. Apparatus comprising: an interferometer comprising a main cavity andan auxiliary reference surface, the main cavity comprising a partiallyreflective surface defining a primary reference surface and a testsurface, the interferometer being configured to direct a primary portionof input electromagnetic radiation to the main cavity and an auxiliaryportion of the input electromagnetic radiation to reflect from theauxiliary reference surface, wherein a first portion of the primaryportion in the main cavity reflects from the primary reference surfaceand a second portion of the primary portion in the main cavity passesthrough the primary reference surface and reflects from the testsurface, the interferometer being further configured to direct theelectromagnetic radiation reflected from the test surface, the primaryreference surface, and the auxiliary reference surface to amulti-element detector to interfere with one another to form aninterference pattern, wherein the auxiliary reference surface is tiltedso that the paths of the electromagnetic radiation reflected from theprimary reference surface and auxiliary reference are non-parallel atthe multi-element detector and the auxiliary reference surface is in thepath of the primary portion of the electromagnetic radiation.
 2. Theapparatus of claim 1, wherein the auxiliary reference surface transmitsthe primary portion of the electromagnetic radiation.
 3. The apparatusof claim 1, wherein the main cavity defines a Fizeau cavity.
 4. Theapparatus of claim 1, wherein there are no beam shaping optics in thebeam path of second portion between the primary reference surface andthe test surface.
 5. An apparatus comprising: an interferometercomprising a main cavity and an auxiliary reference surface, the maincavity comprising a partially reflective surface defining a primaryreference surface and a test surface, the interferometer beingconfigured to direct input electromagnetic radiation to reflect from theauxiliary reference surface, to reflect from the primary referencesurface and to reflect from the test surface, the interferometer beingfurther configured to direct the electromagnetic radiation reflectedfrom the test surface, the primary reference surface, and the auxiliaryreference surface to a multi-element detector to interfere with oneanother to form an interference pattern, wherein the auxiliary referencesurface is configured to introduce spatial carrier fringes in theinterference pattern at the multi-element detector.
 6. The apparatus ofclaim 5, wherein the auxiliary reference surface is positioned in thepath of the input electromagnetic radiation directed to the primaryreference surface.
 7. The apparatus of claim 5, further comprising abeam splitter configured to split the input electromagnetic radiationinto an auxiliary portion and a primary portion, wherein the beamsplitter directs the auxiliary portion along a path to the auxiliaryreference surface and the beam splitter directs the primary portionalong a path to the primary reference surface.
 8. The apparatus of claim5, wherein the auxiliary reference surface is a partially transmissivesurface.
 9. The apparatus of claim 5, wherein the auxiliary referencesurface is a reflective surface.
 10. The apparatus of claim 5, whereinthe auxiliary reference surface is a planar surface.
 11. The apparatusof claim 10, wherein the primary reference surface is a planar surface.12. The apparatus of claim 11, wherein the auxiliary reference surfaceand the primary reference surface are non-parallel.
 13. The apparatus ofclaim 5, wherein the primary reference surface and the auxiliaryreference surface are opposing surfaces of an optical component.
 14. Theapparatus of claim 13, wherein the optical component is a transmissivewedge.
 15. The apparatus of claim 5, wherein the main cavity defines aFizeau cavity.
 16. The apparatus of claim 5, wherein the auxiliaryreference surface is configured to transmit the input electromagneticradiation directed to reflect from the primary reference surface. 17.The apparatus of claim 5, wherein the primary reference surface isconfigured to reflect input electromagnetic radiation along a first pathand the auxiliary reference surface is configured to reflect inputelectromagnetic radiation along a second path, where the second pathoverlaps the first path and is non-parallel to the first path.
 18. Theapparatus of claim 5, wherein there are no beam shaping optics in thebeam path of second portion between the primary reference surface andthe test surface.
 19. The apparatus of claim 5, further comprising ameans for selectively preventing electromagnetic radiation from the testsurface from reaching the detector.
 20. The apparatus of claim 5,further comprising the multi-element detector and an electroniccontroller, wherein the electronic controller is configured to determinesurface profile information about the test surface based on theinterference pattern.
 21. The apparatus of claim 22, wherein theelectronic controller is configured to determine surface profileinformation about the test surface based on the interference patternformed by the electromagnetic radiation reflected from the test surface,the primary reference surface, and the auxiliary reference, and a secondinterference pattern formed by electromagnetic radiation reflected fromthe primary reference surface and the auxiliary reference, with noelectromagnetic radiation reflected from the test surface reaching thedetector.
 22. The apparatus of claim 20, wherein the auxiliary referencesurface is tilted relative to an optical axis of the interferometer toform the spatial carrier fringes in the interference pattern.
 23. Theapparatus of claim 5, further comprising a quadrature phase detectionsystem including the multi-element detector.
 24. The apparatus of claim5, wherein the interferometer comprises a fold optic configured to allowthe primary reference surface to be upward facing and part of a mountconfigured to support the test surface.
 25. The apparatus of claim 5,wherein the auxiliary reference is coupled to a transducer configured tovary an optical path length of electromagnetic radiation reflected fromthe auxiliary reference to the detector.
 26. A system comprising: amulti-element detector; a Fizeau interferometer comprising a primaryreference surface and a test surface, the interferometer beingconfigured to interfere electromagnetic radiation reflected from theprimary reference surface and electromagnetic radiation reflected fromthe test surface to form an interference pattern at the multi-elementdetector; a means for generating spatial carrier fringes in theinterference pattern at the multi-element detector; and an electroniccontroller configured to determine information about the test surfacebased on the interference pattern and the spatial carrier fringes. 27.The system of claim 26, wherein the electronic controller is configuredto determine surface profile information about the test surface based onthe interference pattern and the spatial carrier fringes.
 28. The systemof claim 26, wherein the means for generating the carrier fringescomprises an auxiliary reference mirror configured to direct auxiliaryelectromagnetic radiation along a path to the multi-element detector,wherein the auxiliary electromagnetic radiation interferes with theelectromagnetic radiation reflected from the primary reference surfaceand the test surface to form the carrier fringes in the interferencepattern.